The topology of asymptotically locally flat gravitational instantons

Gábor Etesi

doi:10.1016/j.physletb.2006.08.080

Abstract

In this Letter we demonstrate that the intersection form of the Hausel–Hunsicker–Mazzeo compactification of a four-dimensional ALF gravitational instanton is definite and diagonalizable over the integers if one of the Kähler forms of the hyper-Kähler gravitational instanton metric is exact. This leads to their topological classification. The proof exploits the relationship between L 2 cohomology and U ( 1 ) anti-instantons over gravitational instantons recognized by Hitchin. We then interprete these as reducible points in a singular SU ( 2 ) anti-instanton moduli space over the compactification leading to the identification of its intersection form. This observation on the intersection form might be a useful tool in the full geometric classification of various asymptotically locally flat gravitational instantons.

Highlights

By a gravitational instanton we mean a connected, four dimensional complete hyper-Kahler Riemannian manifold
In case of an ALF gravitational instanton a more intrinsic relationship exists between L2 cohomology and anti-self-duality which removes the disturbing self-dual-anti-self-dual ambiguity from this construction
Taking into account (2), which rules out fractional holonomy of the connection along the disks, and referring to a codimension 2 singularity removal theorem of Sibner and Sibner [18] and Rade [17] one can show that the SU(2) anti-instanton ∇ ⊕ ∇−1 extends over B∞ as a smooth SU(2) connection which is not anti-self-dual with respect to the regularized metric gε

Summary

Introduction

By a gravitational instanton we mean a connected, four dimensional complete hyper-Kahler Riemannian manifold. Since the only compact four dimensional hyper-Kahler spaces up to universal covering are diffeomorphic to the flat torus T 4 or a K3 surface, for further solutions we have to seek non-compact examples Compactness in this case is naturally replaced by the condition that the metric be complete and decay to the flat metric somehow at infinity. Non-compact examples which satisfy the above fall-off conditions are for instance the Euclidean Schwarzschild solution or the Euclidean Kerr–Newman solution which are complete ALF, Ricci flat but not hyper-Kahler spaces [13]. In this letter we focus our attention to the special case of ALF spaces For this class X is a connected, compact, orientable, smooth four-manifold without boundary.

L2 cohomology and anti-self-duality

Topological classification

Concluding remarks